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Theorem List for Metamath Proof Explorer - 22701-22800   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremclmvs1 22701 Scalar product with ring unit. (lmodvs1 18714 analog.) (Contributed by Mario Carneiro, 16-Oct-2015.)
𝑉 = (Base‘𝑊)    &    · = ( ·𝑠𝑊)       ((𝑊 ∈ ℂMod ∧ 𝑋𝑉) → (1 · 𝑋) = 𝑋)

Theoremclmvs2 22702 A vector plus itself is two times the vector. (Contributed by NM, 1-Feb-2007.) (Revised by AV, 21-Sep-2021.)
𝑉 = (Base‘𝑊)    &    · = ( ·𝑠𝑊)    &    + = (+g𝑊)       ((𝑊 ∈ ℂMod ∧ 𝐴𝑉) → (𝐴 + 𝐴) = (2 · 𝐴))

Theoremclm0vs 22703 Zero times a vector is the zero vector. Equation 1a of [Kreyszig] p. 51. (lmod0vs 18719 analog.) (Contributed by Mario Carneiro, 16-Oct-2015.)
𝑉 = (Base‘𝑊)    &   𝐹 = (Scalar‘𝑊)    &    · = ( ·𝑠𝑊)    &    0 = (0g𝑊)       ((𝑊 ∈ ℂMod ∧ 𝑋𝑉) → (0 · 𝑋) = 0 )

Theoremclmopfne 22704 The (functionalized) operations of a complex left module cannot be identical. (Contributed by NM, 31-May-2008.) (Revised by AV, 3-Oct-2021.)
· = ( ·sf𝑊)    &    + = (+𝑓𝑊)       (𝑊 ∈ ℂMod → +· )

Theoremisclmp 22705* The predicate "is a complex left module space." (Contributed by NM, 31-May-2008.) (Revised by AV, 4-Oct-2021.)
· = ( ·𝑠𝑊)    &    + = (+g𝑊)    &   𝑉 = (Base‘𝑊)    &   𝑆 = (Scalar‘𝑊)    &   𝐾 = (Base‘𝑆)       (𝑊 ∈ ℂMod ↔ ((𝑊 ∈ Grp ∧ 𝑆 = (ℂflds 𝐾) ∧ 𝐾 ∈ (SubRing‘ℂfld)) ∧ ∀𝑥𝑉 ((1 · 𝑥) = 𝑥 ∧ ∀𝑦𝐾 ((𝑦 · 𝑥) ∈ 𝑉 ∧ ∀𝑧𝑉 (𝑦 · (𝑥 + 𝑧)) = ((𝑦 · 𝑥) + (𝑦 · 𝑧)) ∧ ∀𝑧𝐾 (((𝑧 + 𝑦) · 𝑥) = ((𝑧 · 𝑥) + (𝑦 · 𝑥)) ∧ ((𝑧 · 𝑦) · 𝑥) = (𝑧 · (𝑦 · 𝑥)))))))

Theoremisclmi0 22706* Properties that determine a complex left module. (Contributed by NM, 5-Nov-2006.) (Revised by AV, 4-Oct-2021.)
· = ( ·𝑠𝑊)    &    + = (+g𝑊)    &   𝑉 = (Base‘𝑊)    &   𝑆 = (Scalar‘𝑊)    &   𝐾 = (Base‘𝑆)    &   𝑆 = (ℂflds 𝐾)    &   𝑊 ∈ Grp    &   𝐾 ∈ (SubRing‘ℂfld)    &   (𝑥𝑉 → (1 · 𝑥) = 𝑥)    &   ((𝑦𝐾𝑥𝑉) → (𝑦 · 𝑥) ∈ 𝑉)    &   ((𝑦𝐾𝑥𝑉𝑧𝑉) → (𝑦 · (𝑥 + 𝑧)) = ((𝑦 · 𝑥) + (𝑦 · 𝑧)))    &   ((𝑦𝐾𝑧𝐾𝑥𝑉) → ((𝑧 + 𝑦) · 𝑥) = ((𝑧 · 𝑥) + (𝑦 · 𝑥)))    &   ((𝑦𝐾𝑧𝐾𝑥𝑉) → ((𝑧 · 𝑦) · 𝑥) = (𝑧 · (𝑦 · 𝑥)))       𝑊 ∈ ℂMod

Theoremclmvneg1 22707 Minus 1 times a vector is the negative of the vector. Equation 2 of [Kreyszig] p. 51. (lmodvneg1 18729 analog.) (Contributed by Mario Carneiro, 16-Oct-2015.)
𝑉 = (Base‘𝑊)    &   𝑁 = (invg𝑊)    &   𝐹 = (Scalar‘𝑊)    &    · = ( ·𝑠𝑊)       ((𝑊 ∈ ℂMod ∧ 𝑋𝑉) → (-1 · 𝑋) = (𝑁𝑋))

Theoremclmvsneg 22708 Multiplication of a vector by a negated scalar. (lmodvsneg 18730 analog.) (Contributed by Mario Carneiro, 16-Oct-2015.)
𝐵 = (Base‘𝑊)    &   𝐹 = (Scalar‘𝑊)    &    · = ( ·𝑠𝑊)    &   𝑁 = (invg𝑊)    &   𝐾 = (Base‘𝐹)    &   (𝜑𝑊 ∈ ℂMod)    &   (𝜑𝑋𝐵)    &   (𝜑𝑅𝐾)       (𝜑 → (𝑁‘(𝑅 · 𝑋)) = (-𝑅 · 𝑋))

Theoremclmmulg 22709 The group multiple function matches the scalar multiplication function. (Contributed by Mario Carneiro, 15-Oct-2015.)
𝑉 = (Base‘𝑊)    &    = (.g𝑊)    &    · = ( ·𝑠𝑊)       ((𝑊 ∈ ℂMod ∧ 𝐴 ∈ ℤ ∧ 𝐵𝑉) → (𝐴 𝐵) = (𝐴 · 𝐵))

Theoremclmsubdir 22710 Scalar multiplication distributive law for subtraction. (lmodsubdir 18744 analog.) (Contributed by Mario Carneiro, 16-Oct-2015.)
𝑉 = (Base‘𝑊)    &    · = ( ·𝑠𝑊)    &   𝐹 = (Scalar‘𝑊)    &   𝐾 = (Base‘𝐹)    &    = (-g𝑊)    &   (𝜑𝑊 ∈ ℂMod)    &   (𝜑𝐴𝐾)    &   (𝜑𝐵𝐾)    &   (𝜑𝑋𝑉)       (𝜑 → ((𝐴𝐵) · 𝑋) = ((𝐴 · 𝑋) (𝐵 · 𝑋)))

Theoremclmpm1dir 22711 Subtractive distributive law for the scalar product of a complex left module. (Contributed by NM, 31-Jul-2007.) (Revised by AV, 21-Sep-2021.)
𝑉 = (Base‘𝑊)    &    · = ( ·𝑠𝑊)    &    + = (+g𝑊)    &   𝐾 = (Base‘(Scalar‘𝑊))       ((𝑊 ∈ ℂMod ∧ (𝐴𝐾𝐵𝐾𝐶𝑉)) → ((𝐴𝐵) · 𝐶) = ((𝐴 · 𝐶) + (-1 · (𝐵 · 𝐶))))

Theoremclmnegneg 22712 Double negative of a vector. (Contributed by NM, 6-Aug-2007.) (Revised by AV, 21-Sep-2021.)
𝑉 = (Base‘𝑊)    &    · = ( ·𝑠𝑊)    &    + = (+g𝑊)       ((𝑊 ∈ ℂMod ∧ 𝐴𝑉) → (-1 · (-1 · 𝐴)) = 𝐴)

Theoremclmnegsubdi2 22713 Distribution of negative over vector subtraction. (Contributed by NM, 6-Aug-2007.) (Revised by AV, 29-Sep-2021.)
𝑉 = (Base‘𝑊)    &    · = ( ·𝑠𝑊)    &    + = (+g𝑊)       ((𝑊 ∈ ℂMod ∧ 𝐴𝑉𝐵𝑉) → (-1 · (𝐴 + (-1 · 𝐵))) = (𝐵 + (-1 · 𝐴)))

Theoremclmsub4 22714 Rearrangement of 4 terms in a mixed vector addition and subtraction. (Contributed by NM, 5-Aug-2007.) (Revised by AV, 29-Sep-2021.)
𝑉 = (Base‘𝑊)    &    · = ( ·𝑠𝑊)    &    + = (+g𝑊)       ((𝑊 ∈ ℂMod ∧ (𝐴𝑉𝐵𝑉) ∧ (𝐶𝑉𝐷𝑉)) → ((𝐴 + 𝐵) + (-1 · (𝐶 + 𝐷))) = ((𝐴 + (-1 · 𝐶)) + (𝐵 + (-1 · 𝐷))))

Theoremclmvsrinv 22715 A vector minus itself. (Contributed by NM, 4-Dec-2006.) (Revised by AV, 28-Sep-2021.)
𝑉 = (Base‘𝑊)    &    · = ( ·𝑠𝑊)    &    + = (+g𝑊)    &    0 = (0g𝑊)       ((𝑊 ∈ ℂMod ∧ 𝐴𝑉) → (𝐴 + (-1 · 𝐴)) = 0 )

Theoremclmvslinv 22716 Minus a vector plus itself. (Contributed by NM, 4-Dec-2006.) (Revised by AV, 28-Sep-2021.)
𝑉 = (Base‘𝑊)    &    · = ( ·𝑠𝑊)    &    + = (+g𝑊)    &    0 = (0g𝑊)       ((𝑊 ∈ ℂMod ∧ 𝐴𝑉) → ((-1 · 𝐴) + 𝐴) = 0 )

Theoremclmvsubval 22717 Value of vector subtraction in terms of addition in a complex module. (lmodvsubval2 18741 analog.) (Contributed by NM, 31-Mar-2014.) (Revised by AV, 7-Oct-2021.)
𝑉 = (Base‘𝑊)    &    + = (+g𝑊)    &    = (-g𝑊)    &   𝐹 = (Scalar‘𝑊)    &    · = ( ·𝑠𝑊)       ((𝑊 ∈ ℂMod ∧ 𝐴𝑉𝐵𝑉) → (𝐴 𝐵) = (𝐴 + (-1 · 𝐵)))

Theoremclmvsubval2 22718 Value of vector subtraction on a complex module. (Contributed by Mario Carneiro, 19-Nov-2013.) (Revised by AV, 7-Oct-2021.)
𝑉 = (Base‘𝑊)    &    + = (+g𝑊)    &    = (-g𝑊)    &   𝐹 = (Scalar‘𝑊)    &    · = ( ·𝑠𝑊)       ((𝑊 ∈ ℂMod ∧ 𝐴𝑉𝐵𝑉) → (𝐴 𝐵) = ((-1 · 𝐵) + 𝐴))

Theoremclmvz 22719 Two ways to express the negative of a vector. (Contributed by NM, 29-Feb-2008.) (Revised by AV, 7-Oct-2021.)
𝑉 = (Base‘𝑊)    &    = (-g𝑊)    &    · = ( ·𝑠𝑊)    &    0 = (0g𝑊)       ((𝑊 ∈ ℂMod ∧ 𝐴𝑉) → ( 0 𝐴) = (-1 · 𝐴))

Theoremzlmclm 22720 The -module operation turns an arbitrary abelian group into a complex module. (Contributed by Mario Carneiro, 30-Oct-2015.)
𝑊 = (ℤMod‘𝐺)       (𝐺 ∈ Abel ↔ 𝑊 ∈ ℂMod)

Theoremclmzlmvsca 22721 The scalar product of a complex module matches the scalar product of the derived -module, which implies, together with zlmbas 19685 and zlmplusg 19686, that any module over is structure-equivalent to the canonical -module ℤMod‘𝐺. (Contributed by Mario Carneiro, 30-Oct-2015.)
𝑊 = (ℤMod‘𝐺)    &   𝑋 = (Base‘𝐺)       ((𝐺 ∈ ℂMod ∧ (𝐴 ∈ ℤ ∧ 𝐵𝑋)) → (𝐴( ·𝑠𝐺)𝐵) = (𝐴( ·𝑠𝑊)𝐵))

Theoremnmoleub2lem 22722* Lemma for nmoleub2a 22725 and similar theorems. (Contributed by Mario Carneiro, 19-Oct-2015.)
𝑁 = (𝑆 normOp 𝑇)    &   𝑉 = (Base‘𝑆)    &   𝐿 = (norm‘𝑆)    &   𝑀 = (norm‘𝑇)    &   𝐺 = (Scalar‘𝑆)    &   𝐾 = (Base‘𝐺)    &   (𝜑𝑆 ∈ (NrmMod ∩ ℂMod))    &   (𝜑𝑇 ∈ (NrmMod ∩ ℂMod))    &   (𝜑𝐹 ∈ (𝑆 LMHom 𝑇))    &   (𝜑𝐴 ∈ ℝ*)    &   (𝜑𝑅 ∈ ℝ+)    &   ((𝜑 ∧ ∀𝑥𝑉 (𝜓 → ((𝑀‘(𝐹𝑥)) / 𝑅) ≤ 𝐴)) → 0 ≤ 𝐴)    &   ((((𝜑 ∧ ∀𝑥𝑉 (𝜓 → ((𝑀‘(𝐹𝑥)) / 𝑅) ≤ 𝐴)) ∧ 𝐴 ∈ ℝ) ∧ (𝑦𝑉𝑦 ≠ (0g𝑆))) → (𝑀‘(𝐹𝑦)) ≤ (𝐴 · (𝐿𝑦)))    &   ((𝜑𝑥𝑉) → (𝜓 → (𝐿𝑥) ≤ 𝑅))       (𝜑 → ((𝑁𝐹) ≤ 𝐴 ↔ ∀𝑥𝑉 (𝜓 → ((𝑀‘(𝐹𝑥)) / 𝑅) ≤ 𝐴)))

Theoremnmoleub2lem3 22723* Lemma for nmoleub2a 22725 and similar theorems. (Contributed by Mario Carneiro, 19-Oct-2015.) (Proof shortened by AV, 29-Sep-2021.)
𝑁 = (𝑆 normOp 𝑇)    &   𝑉 = (Base‘𝑆)    &   𝐿 = (norm‘𝑆)    &   𝑀 = (norm‘𝑇)    &   𝐺 = (Scalar‘𝑆)    &   𝐾 = (Base‘𝐺)    &   (𝜑𝑆 ∈ (NrmMod ∩ ℂMod))    &   (𝜑𝑇 ∈ (NrmMod ∩ ℂMod))    &   (𝜑𝐹 ∈ (𝑆 LMHom 𝑇))    &   (𝜑𝐴 ∈ ℝ*)    &   (𝜑𝑅 ∈ ℝ+)    &   (𝜑 → ℚ ⊆ 𝐾)    &    · = ( ·𝑠𝑆)    &   (𝜑𝐴 ∈ ℝ)    &   (𝜑 → 0 ≤ 𝐴)    &   (𝜑𝐵𝑉)    &   (𝜑𝐵 ≠ (0g𝑆))    &   (𝜑 → ((𝑟 · 𝐵) ∈ 𝑉 → ((𝐿‘(𝑟 · 𝐵)) < 𝑅 → ((𝑀‘(𝐹‘(𝑟 · 𝐵))) / 𝑅) ≤ 𝐴)))    &   (𝜑 → ¬ (𝑀‘(𝐹𝐵)) ≤ (𝐴 · (𝐿𝐵)))        ¬ 𝜑

Theoremnmoleub2lem2 22724* Lemma for nmoleub2a 22725 and similar theorems. (Contributed by Mario Carneiro, 19-Oct-2015.)
𝑁 = (𝑆 normOp 𝑇)    &   𝑉 = (Base‘𝑆)    &   𝐿 = (norm‘𝑆)    &   𝑀 = (norm‘𝑇)    &   𝐺 = (Scalar‘𝑆)    &   𝐾 = (Base‘𝐺)    &   (𝜑𝑆 ∈ (NrmMod ∩ ℂMod))    &   (𝜑𝑇 ∈ (NrmMod ∩ ℂMod))    &   (𝜑𝐹 ∈ (𝑆 LMHom 𝑇))    &   (𝜑𝐴 ∈ ℝ*)    &   (𝜑𝑅 ∈ ℝ+)    &   (𝜑 → ℚ ⊆ 𝐾)    &   (((𝐿𝑥) ∈ ℝ ∧ 𝑅 ∈ ℝ) → ((𝐿𝑥)𝑂𝑅 → (𝐿𝑥) ≤ 𝑅))    &   (((𝐿𝑥) ∈ ℝ ∧ 𝑅 ∈ ℝ) → ((𝐿𝑥) < 𝑅 → (𝐿𝑥)𝑂𝑅))       (𝜑 → ((𝑁𝐹) ≤ 𝐴 ↔ ∀𝑥𝑉 ((𝐿𝑥)𝑂𝑅 → ((𝑀‘(𝐹𝑥)) / 𝑅) ≤ 𝐴)))

Theoremnmoleub2a 22725* The operator norm is the supremum of the value of a linear operator in the closed unit ball. (Contributed by Mario Carneiro, 19-Oct-2015.)
𝑁 = (𝑆 normOp 𝑇)    &   𝑉 = (Base‘𝑆)    &   𝐿 = (norm‘𝑆)    &   𝑀 = (norm‘𝑇)    &   𝐺 = (Scalar‘𝑆)    &   𝐾 = (Base‘𝐺)    &   (𝜑𝑆 ∈ (NrmMod ∩ ℂMod))    &   (𝜑𝑇 ∈ (NrmMod ∩ ℂMod))    &   (𝜑𝐹 ∈ (𝑆 LMHom 𝑇))    &   (𝜑𝐴 ∈ ℝ*)    &   (𝜑𝑅 ∈ ℝ+)    &   (𝜑 → ℚ ⊆ 𝐾)       (𝜑 → ((𝑁𝐹) ≤ 𝐴 ↔ ∀𝑥𝑉 ((𝐿𝑥) ≤ 𝑅 → ((𝑀‘(𝐹𝑥)) / 𝑅) ≤ 𝐴)))

Theoremnmoleub2b 22726* The operator norm is the supremum of the value of a linear operator in the open unit ball. (Contributed by Mario Carneiro, 19-Oct-2015.)
𝑁 = (𝑆 normOp 𝑇)    &   𝑉 = (Base‘𝑆)    &   𝐿 = (norm‘𝑆)    &   𝑀 = (norm‘𝑇)    &   𝐺 = (Scalar‘𝑆)    &   𝐾 = (Base‘𝐺)    &   (𝜑𝑆 ∈ (NrmMod ∩ ℂMod))    &   (𝜑𝑇 ∈ (NrmMod ∩ ℂMod))    &   (𝜑𝐹 ∈ (𝑆 LMHom 𝑇))    &   (𝜑𝐴 ∈ ℝ*)    &   (𝜑𝑅 ∈ ℝ+)    &   (𝜑 → ℚ ⊆ 𝐾)       (𝜑 → ((𝑁𝐹) ≤ 𝐴 ↔ ∀𝑥𝑉 ((𝐿𝑥) < 𝑅 → ((𝑀‘(𝐹𝑥)) / 𝑅) ≤ 𝐴)))

Theoremnmoleub3 22727* The operator norm is the supremum of the value of a linear operator on the closed unit sphere. (Contributed by Mario Carneiro, 19-Oct-2015.) (Proof shortened by AV, 29-Sep-2021.)
𝑁 = (𝑆 normOp 𝑇)    &   𝑉 = (Base‘𝑆)    &   𝐿 = (norm‘𝑆)    &   𝑀 = (norm‘𝑇)    &   𝐺 = (Scalar‘𝑆)    &   𝐾 = (Base‘𝐺)    &   (𝜑𝑆 ∈ (NrmMod ∩ ℂMod))    &   (𝜑𝑇 ∈ (NrmMod ∩ ℂMod))    &   (𝜑𝐹 ∈ (𝑆 LMHom 𝑇))    &   (𝜑𝐴 ∈ ℝ*)    &   (𝜑𝑅 ∈ ℝ+)    &   (𝜑 → 0 ≤ 𝐴)    &   (𝜑 → ℝ ⊆ 𝐾)       (𝜑 → ((𝑁𝐹) ≤ 𝐴 ↔ ∀𝑥𝑉 ((𝐿𝑥) = 𝑅 → ((𝑀‘(𝐹𝑥)) / 𝑅) ≤ 𝐴)))

Theoremnmhmcn 22728 A linear operator over a normed complex module is bounded iff it is continuous. (Contributed by Mario Carneiro, 22-Oct-2015.)
𝐽 = (TopOpen‘𝑆)    &   𝐾 = (TopOpen‘𝑇)    &   𝐺 = (Scalar‘𝑆)    &   𝐵 = (Base‘𝐺)       ((𝑆 ∈ (NrmMod ∩ ℂMod) ∧ 𝑇 ∈ (NrmMod ∩ ℂMod) ∧ ℚ ⊆ 𝐵) → (𝐹 ∈ (𝑆 NMHom 𝑇) ↔ (𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝐹 ∈ (𝐽 Cn 𝐾))))

Theoremcmodscexp 22729 The powers of i belong to the scalar subring of a complex module if i belongs to the scalar subring . (Contributed by AV, 18-Oct-2021.)
𝐹 = (Scalar‘𝑊)    &   𝐾 = (Base‘𝐹)       (((𝑊 ∈ ℂMod ∧ i ∈ 𝐾) ∧ 𝑁 ∈ ℕ) → (i↑𝑁) ∈ 𝐾)

Theoremcmodscmulexp 22730 The scalar product of a vector with powers of i belongs to the base set of a complex module if the scalar subring of th complex module contains i. (Contributed by AV, 18-Oct-2021.)
𝐹 = (Scalar‘𝑊)    &   𝐾 = (Base‘𝐹)    &   𝑋 = (Base‘𝑊)    &    · = ( ·𝑠𝑊)       ((𝑊 ∈ ℂMod ∧ (i ∈ 𝐾𝐵𝑋𝑁 ∈ ℕ)) → ((i↑𝑁) · 𝐵) ∈ 𝑋)

12.5.2  Complex vector spaces

Usually, "complex vector spaces" are vector spaces over the field of the complex numbers, see for example the definition in [Roman] p. 36. In the following, however, the class ℂVec of "complex vector spaces" is generalized to include all vector spaces over subrings of the field of the complex numbers which are division rings (see iscvs 22735). This is analogous to the definition ℂMod of "complex left modules", which are left modules over arbitrary subrings of the field of the complex numbers. Therefore, the real vector spaces (see recvs 22754) and even the rational vector spaces (see qcvs 22755) also belong to ℂVec. So that ZZ-modules (that are the same as Abelian groups, see zlmclm 22720) are complex left modules, but are not complex vector spaces with our definitions (see zclmncvs 22756), because the ring ZZ is not a division subring of CC, see zringndrg 19657.

Syntaxccvs 22731 Complex vector space.
class ℂVec

Definitiondf-cvs 22732 Define a complex vector space, which is just a complex left module and a vector space. (Contributed by Thierry Arnoux, 22-May-2019.)
ℂVec = (ℂMod ∩ LVec)

Theoremcvslvec 22733 A complex vector space is a (left) vector space. (Contributed by Thierry Arnoux, 22-May-2019.)
(𝜑𝑊 ∈ ℂVec)       (𝜑𝑊 ∈ LVec)

Theoremcvsclm 22734 A complex vector space is a complex left module. (Contributed by Thierry Arnoux, 22-May-2019.)
(𝜑𝑊 ∈ ℂVec)       (𝜑𝑊 ∈ ℂMod)

Theoremiscvs 22735 A complex vector space is a complex left module over a division ring. For example, the (complex) left modules over the rational or real or complex numbers are complex vector spaces. (Contributed by AV, 4-Oct-2021.)
(𝑊 ∈ ℂVec ↔ (𝑊 ∈ ℂMod ∧ (Scalar‘𝑊) ∈ DivRing))

Theoremiscvsp 22736* The predicate "is a complex vector space." (Contributed by NM, 31-May-2008.) (Revised by AV, 4-Oct-2021.)
· = ( ·𝑠𝑊)    &    + = (+g𝑊)    &   𝑉 = (Base‘𝑊)    &   𝑆 = (Scalar‘𝑊)    &   𝐾 = (Base‘𝑆)       (𝑊 ∈ ℂVec ↔ ((𝑊 ∈ Grp ∧ (𝑆 ∈ DivRing ∧ 𝑆 = (ℂflds 𝐾)) ∧ 𝐾 ∈ (SubRing‘ℂfld)) ∧ ∀𝑥𝑉 ((1 · 𝑥) = 𝑥 ∧ ∀𝑦𝐾 ((𝑦 · 𝑥) ∈ 𝑉 ∧ ∀𝑧𝑉 (𝑦 · (𝑥 + 𝑧)) = ((𝑦 · 𝑥) + (𝑦 · 𝑧)) ∧ ∀𝑧𝐾 (((𝑧 + 𝑦) · 𝑥) = ((𝑧 · 𝑥) + (𝑦 · 𝑥)) ∧ ((𝑧 · 𝑦) · 𝑥) = (𝑧 · (𝑦 · 𝑥)))))))

Theoremiscvsi 22737* Properties that determine a complex vector space. (Contributed by NM, 5-Nov-2006.) (Revised by AV, 4-Oct-2021.)
· = ( ·𝑠𝑊)    &    + = (+g𝑊)    &   𝑉 = (Base‘𝑊)    &   𝑆 = (Scalar‘𝑊)    &   𝐾 = (Base‘𝑆)    &   𝑊 ∈ Grp    &   𝑆 = (ℂflds 𝐾)    &   𝑆 ∈ DivRing    &   𝐾 ∈ (SubRing‘ℂfld)    &   (𝑥𝑉 → (1 · 𝑥) = 𝑥)    &   ((𝑦𝐾𝑥𝑉) → (𝑦 · 𝑥) ∈ 𝑉)    &   ((𝑦𝐾𝑥𝑉𝑧𝑉) → (𝑦 · (𝑥 + 𝑧)) = ((𝑦 · 𝑥) + (𝑦 · 𝑧)))    &   ((𝑦𝐾𝑧𝐾𝑥𝑉) → ((𝑧 + 𝑦) · 𝑥) = ((𝑧 · 𝑥) + (𝑦 · 𝑥)))    &   ((𝑦𝐾𝑧𝐾𝑥𝑉) → ((𝑧 · 𝑦) · 𝑥) = (𝑧 · (𝑦 · 𝑥)))       𝑊 ∈ ℂVec

Theoremcvsi 22738* The properties of a complex vector space, which is an Abelian group (i.e. the vectors, with the operation of vector addition) accompanied by a scalar multiplication operation on the field of complex numbers. (Contributed by NM, 3-Nov-2006.) (Revised by AV, 21-Sep-2021.)
𝑋 = (Base‘𝑊)    &    + = (+g𝑊)    &   𝑆 = (Base‘(Scalar‘𝑊))    &    = ( ·sf𝑊)    &    · = ( ·𝑠𝑊)       (𝑊 ∈ ℂVec → (𝑊 ∈ Abel ∧ (𝑆 ⊆ ℂ ∧ :(𝑆 × 𝑋)⟶𝑋) ∧ ∀𝑥𝑋 ((1 · 𝑥) = 𝑥 ∧ ∀𝑦𝑆 (∀𝑧𝑋 (𝑦 · (𝑥 + 𝑧)) = ((𝑦 · 𝑥) + (𝑦 · 𝑧)) ∧ ∀𝑧𝑆 (((𝑦 + 𝑧) · 𝑥) = ((𝑦 · 𝑥) + (𝑧 · 𝑥)) ∧ ((𝑦 · 𝑧) · 𝑥) = (𝑦 · (𝑧 · 𝑥)))))))

Theoremcvsunit 22739 Unit group of the scalar ring of a complex vector space. (Contributed by Thierry Arnoux, 22-May-2019.)
𝐹 = (Scalar‘𝑊)    &   𝐾 = (Base‘𝐹)       (𝑊 ∈ ℂVec → (𝐾 ∖ {0}) = (Unit‘𝐹))

Theoremcvsdiv 22740 Division of the scalar ring of a complex vector space. (Contributed by Thierry Arnoux, 22-May-2019.)
𝐹 = (Scalar‘𝑊)    &   𝐾 = (Base‘𝐹)       ((𝑊 ∈ ℂVec ∧ (𝐴𝐾𝐵𝐾𝐵 ≠ 0)) → (𝐴 / 𝐵) = (𝐴(/r𝐹)𝐵))

Theoremcvsdivcl 22741 The scalar field of a complex vector space is closed under division. (Contributed by Thierry Arnoux, 22-May-2019.)
𝐹 = (Scalar‘𝑊)    &   𝐾 = (Base‘𝐹)       ((𝑊 ∈ ℂVec ∧ (𝐴𝐾𝐵𝐾𝐵 ≠ 0)) → (𝐴 / 𝐵) ∈ 𝐾)

Theoremcvsmuleqdivd 22742 An equality involving ratios in a complex vector space. (Contributed by Thierry Arnoux, 22-May-2019.)
𝑉 = (Base‘𝑊)    &    · = ( ·𝑠𝑊)    &   𝐹 = (Scalar‘𝑊)    &   𝐾 = (Base‘𝐹)    &   (𝜑𝑊 ∈ ℂVec)    &   (𝜑𝐴𝐾)    &   (𝜑𝐵𝐾)    &   (𝜑𝑋𝑉)    &   (𝜑𝑌𝑉)    &   (𝜑𝐴 ≠ 0)    &   (𝜑 → (𝐴 · 𝑋) = (𝐵 · 𝑌))       (𝜑𝑋 = ((𝐵 / 𝐴) · 𝑌))

Theoremcvsdiveqd 22743 An equality involving ratios in a complex vector space. (Contributed by Thierry Arnoux, 22-May-2019.)
𝑉 = (Base‘𝑊)    &    · = ( ·𝑠𝑊)    &   𝐹 = (Scalar‘𝑊)    &   𝐾 = (Base‘𝐹)    &   (𝜑𝑊 ∈ ℂVec)    &   (𝜑𝐴𝐾)    &   (𝜑𝐵𝐾)    &   (𝜑𝑋𝑉)    &   (𝜑𝑌𝑉)    &   (𝜑𝐴 ≠ 0)    &   (𝜑𝐵 ≠ 0)    &   (𝜑𝑋 = ((𝐴 / 𝐵) · 𝑌))       (𝜑 → ((𝐵 / 𝐴) · 𝑋) = 𝑌)

Theoremcnlmodlem1 22744 Lemma 1 for cnlmod 22748. (Contributed by AV, 20-Sep-2021.)
𝑊 = ({⟨(Base‘ndx), ℂ⟩, ⟨(+g‘ndx), + ⟩} ∪ {⟨(Scalar‘ndx), ℂfld⟩, ⟨( ·𝑠 ‘ndx), · ⟩})       (Base‘𝑊) = ℂ

Theoremcnlmodlem2 22745 Lemma 2 for cnlmod 22748. (Contributed by AV, 20-Sep-2021.)
𝑊 = ({⟨(Base‘ndx), ℂ⟩, ⟨(+g‘ndx), + ⟩} ∪ {⟨(Scalar‘ndx), ℂfld⟩, ⟨( ·𝑠 ‘ndx), · ⟩})       (+g𝑊) = +

Theoremcnlmodlem3 22746 Lemma 3 for cnlmod 22748. (Contributed by AV, 20-Sep-2021.)
𝑊 = ({⟨(Base‘ndx), ℂ⟩, ⟨(+g‘ndx), + ⟩} ∪ {⟨(Scalar‘ndx), ℂfld⟩, ⟨( ·𝑠 ‘ndx), · ⟩})       (Scalar‘𝑊) = ℂfld

Theoremcnlmod4 22747 Lemma 4 for cnlmod 22748. (Contributed by AV, 20-Sep-2021.)
𝑊 = ({⟨(Base‘ndx), ℂ⟩, ⟨(+g‘ndx), + ⟩} ∪ {⟨(Scalar‘ndx), ℂfld⟩, ⟨( ·𝑠 ‘ndx), · ⟩})       ( ·𝑠𝑊) = ·

Theoremcnlmod 22748 The set of complex numbers is a left module over itself. The vector operation is +, and the scalar product is ·. (Contributed by AV, 20-Sep-2021.)
𝑊 = ({⟨(Base‘ndx), ℂ⟩, ⟨(+g‘ndx), + ⟩} ∪ {⟨(Scalar‘ndx), ℂfld⟩, ⟨( ·𝑠 ‘ndx), · ⟩})       𝑊 ∈ LMod

Theoremcnstrcvs 22749 The set of complex numbers is a complex vector space. The vector operation is +, and the scalar product is ·. (Contributed by NM, 5-Nov-2006.) (Revised by AV, 20-Sep-2021.)
𝑊 = ({⟨(Base‘ndx), ℂ⟩, ⟨(+g‘ndx), + ⟩} ∪ {⟨(Scalar‘ndx), ℂfld⟩, ⟨( ·𝑠 ‘ndx), · ⟩})       𝑊 ∈ ℂVec

Theoremcnrbas 22750 The set of complex numbers is the base set of the complex numbers as left module induced by the complex numbers as subring of itself. (Contributed by AV, 21-Sep-2021.)
𝐶 = (ringLMod‘ℂfld)       (Base‘𝐶) = ℂ

Theoremcnrlmod 22751 The set of complex numbers (as a subring of itself) is a left module over itself. The vector operation is +, and the scalar product is ·. (Contributed by AV, 21-Sep-2021.)
𝐶 = (ringLMod‘ℂfld)       𝐶 ∈ LMod

Theoremcnrlvec 22752 The set of complex numbers (as a subring of itself) is a (left) vector space over itself. The vector operation is +, and the scalar product is ·. (Contributed by AV, 21-Sep-2021.)
𝐶 = (ringLMod‘ℂfld)       𝐶 ∈ LVec

Theoremcncvs 22753 The set of complex numbers (as a subring of itself) is a complex vector space. The vector operation is +, and the scalar product is ·. (Contributed by NM, 5-Nov-2006.) (Revised by AV, 21-Sep-2021.)
𝐶 = (ringLMod‘ℂfld)       𝐶 ∈ ℂVec

Theoremrecvs 22754 The set of real numbers (as a subring of itself) is a complex vector space. The vector operation is +, and the scalar product is ·. (Contributed by AV, 22-Oct-2021.)
𝑅 = (ringLMod‘ℝfld)       𝑅 ∈ ℂVec

Theoremqcvs 22755 The set of rational numbers (as a subring of itself) is a complex vector space. The vector operation is +, and the scalar product is ·. (Contributed by AV, 22-Oct-2021.)
𝑄 = (ringLMod‘(ℂflds ℚ))       𝑄 ∈ ℂVec

Theoremzclmncvs 22756 The set of integers (as a subring of itself) is a complex module, but not a complex vector space. The vector operation is +, and the scalar product is ·. (Contributed by AV, 22-Oct-2021.)
𝑍 = (ringLMod‘ℤring)       (𝑍 ∈ ℂMod ∧ 𝑍 ∉ ℂVec)

12.5.3  Normed complex vector spaces

Until now, there is no need for a definition of normed complex vector spaces. The idiom 𝑊 ∈ (NrmVec ∩ ℂVec) is used in the following to say that 𝑊 is a normed complex vector space, i.e. a complex vector space which is also a normed vector space. The label of the theorems for normed complex vector spaces contain the fragment ncvs.

Theoremisncvsngp 22757* The predicate "is a normed complex vector space." if the underlying group is a normed group. (Contributed by NM, 5-Jun-2008.) (Revised by AV, 7-Oct-2021.)
𝑉 = (Base‘𝑊)    &   𝑁 = (norm‘𝑊)    &    · = ( ·𝑠𝑊)    &   𝐹 = (Scalar‘𝑊)    &   𝐾 = (Base‘𝐹)       (𝑊 ∈ (NrmVec ∩ ℂVec) ↔ (𝑊 ∈ ℂVec ∧ 𝑊 ∈ NrmGrp ∧ ∀𝑥𝑉𝑘𝐾 (𝑁‘(𝑘 · 𝑥)) = ((abs‘𝑘) · (𝑁𝑥))))

Theoremisncvsngpd 22758* Properties that determine a normed complex vector space. (Contributed by NM, 15-Apr-2007.) (Revised by AV, 7-Oct-2021.)
𝑉 = (Base‘𝑊)    &   𝑁 = (norm‘𝑊)    &    · = ( ·𝑠𝑊)    &   𝐹 = (Scalar‘𝑊)    &   𝐾 = (Base‘𝐹)    &   (𝜑𝑊 ∈ ℂVec)    &   (𝜑𝑊 ∈ NrmGrp)    &   ((𝜑 ∧ (𝑥𝑉𝑘𝐾)) → (𝑁‘(𝑘 · 𝑥)) = ((abs‘𝑘) · (𝑁𝑥)))       (𝜑𝑊 ∈ (NrmVec ∩ ℂVec))

Theoremncvsi 22759* The properties of a normed complex vector space, which is a vector space accompanied by a norm. (Contributed by NM, 11-Nov-2006.) (Revised by AV, 7-Oct-2021.)
𝑉 = (Base‘𝑊)    &   𝑁 = (norm‘𝑊)    &    · = ( ·𝑠𝑊)    &   𝐹 = (Scalar‘𝑊)    &   𝐾 = (Base‘𝐹)    &    = (-g𝑊)    &    0 = (0g𝑊)       (𝑊 ∈ (NrmVec ∩ ℂVec) → (𝑊 ∈ ℂVec ∧ 𝑁:𝑉⟶ℝ ∧ ∀𝑥𝑉 (((𝑁𝑥) = 0 ↔ 𝑥 = 0 ) ∧ ∀𝑦𝑉 (𝑁‘(𝑥 𝑦)) ≤ ((𝑁𝑥) + (𝑁𝑦)) ∧ ∀𝑘𝐾 (𝑁‘(𝑘 · 𝑥)) = ((abs‘𝑘) · (𝑁𝑥)))))

Theoremncvsprp 22760 Proportionality property of the norm of a scalar product in a normed complex vector space. (Contributed by NM, 11-Nov-2006.) (Revised by AV, 8-Oct-2021.)
𝑉 = (Base‘𝑊)    &   𝑁 = (norm‘𝑊)    &    · = ( ·𝑠𝑊)    &   𝐹 = (Scalar‘𝑊)    &   𝐾 = (Base‘𝐹)       ((𝑊 ∈ (NrmVec ∩ ℂVec) ∧ 𝐴𝐾𝐵𝑉) → (𝑁‘(𝐴 · 𝐵)) = ((abs‘𝐴) · (𝑁𝐵)))

Theoremncvsge0 22761 The norm of a scalar product with a nonnegative real. (Contributed by NM, 1-Jan-2008.) (Revised by AV, 8-Oct-2021.)
𝑉 = (Base‘𝑊)    &   𝑁 = (norm‘𝑊)    &    · = ( ·𝑠𝑊)    &   𝐹 = (Scalar‘𝑊)    &   𝐾 = (Base‘𝐹)       ((𝑊 ∈ (NrmVec ∩ ℂVec) ∧ (𝐴 ∈ (𝐾 ∩ ℝ) ∧ 0 ≤ 𝐴) ∧ 𝐵𝑉) → (𝑁‘(𝐴 · 𝐵)) = (𝐴 · (𝑁𝐵)))

Theoremncvsm1 22762 The norm of the negative of a vector. (Contributed by NM, 28-Nov-2006.) (Revised by AV, 8-Oct-2021.)
𝑉 = (Base‘𝑊)    &   𝑁 = (norm‘𝑊)    &    · = ( ·𝑠𝑊)       ((𝑊 ∈ (NrmVec ∩ ℂVec) ∧ 𝐴𝑉) → (𝑁‘(-1 · 𝐴)) = (𝑁𝐴))

Theoremncvsdif 22763 The norm of the difference between two vectors. (Contributed by NM, 1-Dec-2006.) (Revised by AV, 8-Oct-2021.)
𝑉 = (Base‘𝑊)    &   𝑁 = (norm‘𝑊)    &    · = ( ·𝑠𝑊)    &    + = (+g𝑊)       ((𝑊 ∈ (NrmVec ∩ ℂVec) ∧ 𝐴𝑉𝐵𝑉) → (𝑁‘(𝐴 + (-1 · 𝐵))) = (𝑁‘(𝐵 + (-1 · 𝐴))))

Theoremncvspi 22764 The norm of a vector plus the imaginary scalar product of another. (Contributed by NM, 2-Feb-2007.) (Revised by AV, 8-Oct-2021.)
𝑉 = (Base‘𝑊)    &   𝑁 = (norm‘𝑊)    &    · = ( ·𝑠𝑊)    &    + = (+g𝑊)    &   𝐹 = (Scalar‘𝑊)    &   𝐾 = (Base‘𝐹)       ((𝑊 ∈ (NrmVec ∩ ℂVec) ∧ (𝐴𝑉𝐵𝑉) ∧ i ∈ 𝐾) → (𝑁‘(𝐴 + (i · 𝐵))) = (𝑁‘(𝐵 + (-i · 𝐴))))

Theoremncvs1 22765 From any nonzero vector, construct a vector whose norm is one. (Contributed by NM, 6-Dec-2007.) (Revised by AV, 8-Oct-2021.)
𝑋 = (Base‘𝐺)    &   𝑁 = (norm‘𝐺)    &    0 = (0g𝐺)    &    · = ( ·𝑠𝐺)    &   𝐹 = (Scalar‘𝐺)    &   𝐾 = (Base‘𝐹)       ((𝐺 ∈ (NrmVec ∩ ℂVec) ∧ (𝐴𝑋𝐴0 ) ∧ (1 / (𝑁𝐴)) ∈ 𝐾) → (𝑁‘((1 / (𝑁𝐴)) · 𝐴)) = 1)

Theoremcnrnvc 22766 The set of complex numbers (as a subring of itself) is a normed vector space over itself. The vector operation is +, and the scalar product is ·, and the norm function is abs. (Contributed by AV, 9-Oct-2021.)
𝐶 = (ringLMod‘ℂfld)       𝐶 ∈ NrmVec

Theoremcnncvs 22767 The set of complex numbers (as a subring of itself) is a normed complex vector space. The vector operation is +, and the scalar product is ·, and the norm function is abs. (Contributed by Steve Rodriguez, 3-Dec-2006.) (Revised by AV, 9-Oct-2021.)
𝐶 = (ringLMod‘ℂfld)       𝐶 ∈ (NrmVec ∩ ℂVec)

Theoremcnnm 22768 The norm operation of the normed complex vector space of complex numbers. (Contributed by NM, 12-Jan-2008.) (Revised by AV, 9-Oct-2021.)
𝐶 = (ringLMod‘ℂfld)       (norm‘𝐶) = abs

Theoremncvspds 22769 Value of the distance function in terms of the norm of a normed complex vector space. Equation 1 of [Kreyszig] p. 59. (Contributed by NM, 28-Nov-2006.) (Revised by AV, 13-Oct-2021.)
𝑁 = (norm‘𝐺)    &   𝑋 = (Base‘𝐺)    &    + = (+g𝐺)    &   𝐷 = (dist‘𝐺)    &    · = ( ·𝑠𝐺)       ((𝐺 ∈ (NrmVec ∩ ℂVec) ∧ 𝐴𝑋𝐵𝑋) → (𝐴𝐷𝐵) = (𝑁‘(𝐴 + (-1 · 𝐵))))

Theoremcnindmet 22770 The metric induced on the complex numbers. cnmet 22385 proves that it is a metric. The induced metric is identical with the original metric on the complex numbers, see cnfldds 19577 and also cnmet 22385. (Contributed by Steve Rodriguez, 5-Dec-2006.) (Revised by AV, 17-Oct-2021.)
𝑇 = (ℂfld toNrmGrp abs)       (dist‘𝑇) = (abs ∘ − )

Theoremcnncvsaddassdemo 22771 Derive the associative law for complex number addition addass 9902 to demonstrate the use of the properties of a (normed complex) vector space for the complex numbers. (Contributed by NM, 12-Jan-2008.) (Revised by AV, 9-Oct-2021.) (Proof modification is discouraged.)
((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → ((𝐴 + 𝐵) + 𝐶) = (𝐴 + (𝐵 + 𝐶)))

Theoremcnncvsmulassdemo 22772 Derive the associative law for complex number multiplication mulass 9903 interpreted as scalar multiplication to demonstrate the use of the properties of a (normed) complex vector space for the complex numbers. (Contributed by AV, 9-Oct-2021.) (Proof modification is discouraged.)
((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → ((𝐴 · 𝐵) · 𝐶) = (𝐴 · (𝐵 · 𝐶)))

Theoremcnncvsabsnegdemo 22773 Derive the absolute value of a negative complex number absneg 13865 to demonstrate the use of the properties of a normed complex vector space for the complex numbers. (Contributed by AV, 9-Oct-2021.) (Proof modification is discouraged.)
(𝐴 ∈ ℂ → (abs‘-𝐴) = (abs‘𝐴))

12.5.4  Complex pre-Hilbert space

Syntaxccph 22774 Extend class notation with a complex pre-Hilbert space.
class ℂPreHil

Syntaxctch 22775 Function to put a norm on a Hilbert space.
class toℂHil

Definitiondf-cph 22776* Define a complex pre-Hilbert space. By restricting the scalar field to a quadratically closed subfield of , we have enough structure to define a norm, with the associated connection to a metric and topology. (Contributed by Mario Carneiro, 8-Oct-2015.)
ℂPreHil = {𝑤 ∈ (PreHil ∩ NrmMod) ∣ [(Scalar‘𝑤) / 𝑓][(Base‘𝑓) / 𝑘](𝑓 = (ℂflds 𝑘) ∧ (√ “ (𝑘 ∩ (0[,)+∞))) ⊆ 𝑘 ∧ (norm‘𝑤) = (𝑥 ∈ (Base‘𝑤) ↦ (√‘(𝑥(·𝑖𝑤)𝑥))))}

Definitiondf-tch 22777* Define a function to augment a (pre-)Hilbert space with a norm. No extra parameters are needed, but some conditions must be satisfied to ensure that this in fact creates a normed pre-Hilbert space. (Contributed by Mario Carneiro, 7-Oct-2015.)
toℂHil = (𝑤 ∈ V ↦ (𝑤 toNrmGrp (𝑥 ∈ (Base‘𝑤) ↦ (√‘(𝑥(·𝑖𝑤)𝑥)))))

Theoremiscph 22778* A complex pre-Hilbert space is a pre-Hilbert space over a quadratically closed subfield of the complex numbers, with a norm defined. (Contributed by Mario Carneiro, 8-Oct-2015.)
𝑉 = (Base‘𝑊)    &    , = (·𝑖𝑊)    &   𝑁 = (norm‘𝑊)    &   𝐹 = (Scalar‘𝑊)    &   𝐾 = (Base‘𝐹)       (𝑊 ∈ ℂPreHil ↔ ((𝑊 ∈ PreHil ∧ 𝑊 ∈ NrmMod ∧ 𝐹 = (ℂflds 𝐾)) ∧ (√ “ (𝐾 ∩ (0[,)+∞))) ⊆ 𝐾𝑁 = (𝑥𝑉 ↦ (√‘(𝑥 , 𝑥)))))

Theoremcphphl 22779 A complex pre-Hilbert space is a pre-Hilbert space. (Contributed by Mario Carneiro, 7-Oct-2015.)
(𝑊 ∈ ℂPreHil → 𝑊 ∈ PreHil)

Theoremcphnlm 22780 A complex pre-Hilbert space is a normed module. (Contributed by Mario Carneiro, 7-Oct-2015.)
(𝑊 ∈ ℂPreHil → 𝑊 ∈ NrmMod)

Theoremcphngp 22781 A complex pre-Hilbert space is a normed group. (Contributed by Mario Carneiro, 13-Oct-2015.)
(𝑊 ∈ ℂPreHil → 𝑊 ∈ NrmGrp)

Theoremcphlmod 22782 A complex pre-Hilbert space is a left module. (Contributed by Mario Carneiro, 7-Oct-2015.)
(𝑊 ∈ ℂPreHil → 𝑊 ∈ LMod)

Theoremcphlvec 22783 A complex pre-Hilbert space is a left vector space. (Contributed by Mario Carneiro, 7-Oct-2015.)
(𝑊 ∈ ℂPreHil → 𝑊 ∈ LVec)

Theoremcphnvc 22784 A complex pre-Hilbert space is a normed vector space. (Contributed by Mario Carneiro, 8-Oct-2015.)
(𝑊 ∈ ℂPreHil → 𝑊 ∈ NrmVec)

Theoremcphsubrglem 22785 Lemma for cphsubrg 22788. (Contributed by Mario Carneiro, 9-Oct-2015.)
𝐾 = (Base‘𝐹)    &   (𝜑𝐹 = (ℂflds 𝐴))    &   (𝜑𝐹 ∈ DivRing)       (𝜑 → (𝐹 = (ℂflds 𝐾) ∧ 𝐾 = (𝐴 ∩ ℂ) ∧ 𝐾 ∈ (SubRing‘ℂfld)))

Theoremcphreccllem 22786 Lemma for cphreccl 22789. (Contributed by Mario Carneiro, 8-Oct-2015.)
𝐾 = (Base‘𝐹)    &   (𝜑𝐹 = (ℂflds 𝐴))    &   (𝜑𝐹 ∈ DivRing)       ((𝜑𝑋𝐾𝑋 ≠ 0) → (1 / 𝑋) ∈ 𝐾)

Theoremcphsca 22787 A complex pre-Hilbert space is a vector space over a subfield of . (Contributed by Mario Carneiro, 8-Oct-2015.)
𝐹 = (Scalar‘𝑊)    &   𝐾 = (Base‘𝐹)       (𝑊 ∈ ℂPreHil → 𝐹 = (ℂflds 𝐾))

Theoremcphsubrg 22788 The scalar field of a complex pre-Hilbert space is a subring of . (Contributed by Mario Carneiro, 8-Oct-2015.)
𝐹 = (Scalar‘𝑊)    &   𝐾 = (Base‘𝐹)       (𝑊 ∈ ℂPreHil → 𝐾 ∈ (SubRing‘ℂfld))

Theoremcphreccl 22789 The scalar field of a complex pre-Hilbert space is closed under reciprocal. (Contributed by Mario Carneiro, 8-Oct-2015.)
𝐹 = (Scalar‘𝑊)    &   𝐾 = (Base‘𝐹)       ((𝑊 ∈ ℂPreHil ∧ 𝐴𝐾𝐴 ≠ 0) → (1 / 𝐴) ∈ 𝐾)

Theoremcphdivcl 22790 The scalar field of a complex pre-Hilbert space is closed under reciprocal. (Contributed by Mario Carneiro, 11-Oct-2015.)
𝐹 = (Scalar‘𝑊)    &   𝐾 = (Base‘𝐹)       ((𝑊 ∈ ℂPreHil ∧ (𝐴𝐾𝐵𝐾𝐵 ≠ 0)) → (𝐴 / 𝐵) ∈ 𝐾)

Theoremcphcjcl 22791 The scalar field of a complex pre-Hilbert space is closed under conjugation. (Contributed by Mario Carneiro, 11-Oct-2015.)
𝐹 = (Scalar‘𝑊)    &   𝐾 = (Base‘𝐹)       ((𝑊 ∈ ℂPreHil ∧ 𝐴𝐾) → (∗‘𝐴) ∈ 𝐾)

Theoremcphsqrtcl 22792 The scalar field of a complex pre-Hilbert space is closed under square roots of positive reals (i.e. it is quadratically closed relative to ). (Contributed by Mario Carneiro, 8-Oct-2015.)
𝐹 = (Scalar‘𝑊)    &   𝐾 = (Base‘𝐹)       ((𝑊 ∈ ℂPreHil ∧ (𝐴𝐾𝐴 ∈ ℝ ∧ 0 ≤ 𝐴)) → (√‘𝐴) ∈ 𝐾)

Theoremcphabscl 22793 The scalar field of a complex pre-Hilbert space is closed under the absolute value operation. (Contributed by Mario Carneiro, 11-Oct-2015.)
𝐹 = (Scalar‘𝑊)    &   𝐾 = (Base‘𝐹)       ((𝑊 ∈ ℂPreHil ∧ 𝐴𝐾) → (abs‘𝐴) ∈ 𝐾)

Theoremcphsqrtcl2 22794 The scalar field of a complex pre-Hilbert space is closed under square roots of all numbers except possibly the negative reals. (Contributed by Mario Carneiro, 8-Oct-2015.)
𝐹 = (Scalar‘𝑊)    &   𝐾 = (Base‘𝐹)       ((𝑊 ∈ ℂPreHil ∧ 𝐴𝐾 ∧ ¬ -𝐴 ∈ ℝ+) → (√‘𝐴) ∈ 𝐾)

Theoremcphsqrtcl3 22795 If the scalar field contains i, it is completely closed under square roots (i.e. it is quadratically closed). (Contributed by Mario Carneiro, 11-Oct-2015.)
𝐹 = (Scalar‘𝑊)    &   𝐾 = (Base‘𝐹)       ((𝑊 ∈ ℂPreHil ∧ i ∈ 𝐾𝐴𝐾) → (√‘𝐴) ∈ 𝐾)

Theoremcphqss 22796 The scalar field of a complex pre-Hilbert space contains all rational numbers. (Contributed by Mario Carneiro, 15-Oct-2015.)
𝐹 = (Scalar‘𝑊)    &   𝐾 = (Base‘𝐹)       (𝑊 ∈ ℂPreHil → ℚ ⊆ 𝐾)

Theoremcphclm 22797 A complex pre-Hilbert space is a complex module. (Contributed by Mario Carneiro, 16-Oct-2015.)
(𝑊 ∈ ℂPreHil → 𝑊 ∈ ℂMod)

Theoremcphnmvs 22798 Norm of a scalar product. (Contributed by Mario Carneiro, 16-Oct-2015.)
𝑉 = (Base‘𝑊)    &   𝑁 = (norm‘𝑊)    &    · = ( ·𝑠𝑊)    &   𝐹 = (Scalar‘𝑊)    &   𝐾 = (Base‘𝐹)       ((𝑊 ∈ ℂPreHil ∧ 𝑋𝐾𝑌𝑉) → (𝑁‘(𝑋 · 𝑌)) = ((abs‘𝑋) · (𝑁𝑌)))

Theoremcphipcl 22799 An inner product is a member of the complex numbers. (Contributed by Mario Carneiro, 13-Oct-2015.)
𝑉 = (Base‘𝑊)    &    , = (·𝑖𝑊)       ((𝑊 ∈ ℂPreHil ∧ 𝐴𝑉𝐵𝑉) → (𝐴 , 𝐵) ∈ ℂ)

Theoremcphnmfval 22800* The value of the norm in a complex pre-Hilbert space is the square root of the inner product of a vector with itself. (Contributed by Mario Carneiro, 7-Oct-2015.)
𝑉 = (Base‘𝑊)    &    , = (·𝑖𝑊)    &   𝑁 = (norm‘𝑊)       (𝑊 ∈ ℂPreHil → 𝑁 = (𝑥𝑉 ↦ (√‘(𝑥 , 𝑥))))

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